CN105676903A - Nonprobability reliability optimization based design method of vibration optimal control system - Google Patents

Abstract

The invention relates to a nonprobability reliability optimization based design method of a vibration optimal control system. According to the method, a finite element equation of structural vibration is converted into a state space form via variable substitution, and a state space control equation of structural vibration is established; a nonprobability reliability analysis theory of an active control system of structural vibration is provided; and based on the provided nonprobability reliability analysis theory, reliability optimization is carried out on a controller obtained via an optimal control theory, and a closed-loop controller satisfying the reliability index is obtained. The uncertainly of a closed-loop control system is overcome in the aspect of reliability, and the problems that optimal control cannot satisfy the reliability requirement and robust control is too conservative can be effectively solved.

Description

A kind of vibration optimal control system method for designing optimized based on Multidisciplinary systems

Technical field

The present invention relates to the technical field of active control in structural vibration, be specifically related to a kind of optimum control weighting function system of selection optimized based on Multidisciplinary systems.

Background technology

Along with the development of China's aeronautical and space technology, the requirement of aerospace equipment structural behaviour is also more and more higher. Having been able under the premise met design requirement at structural static strength, the requirement of structural vibration is also increasingly strict. Particularly in space industry, flexible structure Low rigidity, maximization become an important development trend. Such as space expandable type antenna, spacecraft flexible mechanical arm, solar energy sailboard and its supporting construction. These flexible structures due in space structural damping little, lasting vibratory response will be produced once be disturbed, and needs the die-away time grown very much. These lasting structural vibrations can bring various problem, such as kinematic accuracy not, the problem such as structural fatigue or resonance. Due to the high cost of aerospace equipment, if these structures of clocking requirement can be run in high precision in the time of regulation, and the interference in the external world can not be subject to. It is thus desirable to the vibration of these flexible structures is controlled. Passive control needs to increase vibration isolation or energy-dissipating device to increase the damping of structure, this method simply effectively, be easily achieved, good economy performance and need not be extra energy input. But this passive control methods is poor to low-frequency vibration effect, and owing to adding vibration isolation or energy-dissipating device, the quality of structure inevitably increases, and this leverages the performance of spacecraft. Reducing the quality of spacecraft structure, increase the eternal pursuit that effective mass is engineer, therefore traditional passive control cannot meet design requirement, and Active Vibration Control becomes one of focus of research at present.

Active Vibration Control is into a kind of vibration control method fast-developing over 30 years, and particular with the development of intellectual material, Active Vibration Control technology increasingly receives the concern of people. The 1950's, modern control theory obtains great breakthrough development and innovation, and this also provides theoretical basis for Active Vibration Control. Active Vibration Control is to control power by artificial introducing secondary to make structure occur secondary vibration to superpose with structure initial vibration, is finally reached the purpose eliminating structural vibration. And the control power of this artificial increase is the vibration signal by measurement structure, and inputting a signal in controller, thus output control power in the controller, therefore the design of controller just seems increasingly important. Usually, the mathematical model of system and real system also exist the difference of the aspect such as parameter or structure, and the control law designed is all based on the mathematical model of system mostly, in order to ensure that real system disturbs to external world, the uncertainty etc. of system has sensitivity little as far as possible, result in the research of system kinds of robust control problems. Numerous scholars is being devoted to this research of robustness within very long a period of time, to early 1980s, also achieves very big breakthrough in a lot of fields, and wherein most typical is exactly H_∞Robust control method.But this method is overly conservative to probabilistic estimation, thus adding the energy output of system. For uncertain problem, reliability design approach is a kind of effective solution, it is possible to effectively solve problem overly conservative in robust control. And for the closed-loop control system of additive method design, have the problem that there is poor reliability. The present invention is exactly the angle from Multidisciplinary systems, based on a kind of Method of Active Vibration Control that optimum control proposes. Existing patent documentation and non-patent literature are all without the report of correlation technique.

Summary of the invention

The technical problem to be solved in the present invention is: overcome the deficiency of existing technology, it is provided that a kind of vibration optimal control system method for designing optimized based on Multidisciplinary systems, thus improving the reliability of active control system.

The technology of the present invention solution: a kind of optimum control weighting function system of selection optimized based on Multidisciplinary systems, first with state-space model, the FEM (finite element) model of structural vibration is changed, based on the state-space model after conversion, for the uncertainty that system exists, carry out non-probability analysis, obtain the interval range of system response. Set up the reliability index calculating method of active control system, by the various uncertainties in non-probability interval collection approach quantitative model, analyze the reliability of closed-loop control system. On the basis of fail-safe analysis, set up the reliability optimization model of weighting function, weighting function is optimized, thus controller that is that be met Reliability Constraint and that make control power minimum. Finally the controller after optimization is applied in system, builds vibration Reliable Control Systems.

The technical solution used in the present invention is: a kind of vibration optimal control system method for designing optimized based on Multidisciplinary systems, and the method step is as follows:

The first step: according to Structural Dynamics finite element equation, set up the state space equation of active control system:

$M\stackrel{\·\·}{x}+P\stackrel{\·}{x}+Kx=B_{c}u---\left(1\right)$

Wherein M to be architecture quality matrix, P be structural damping matrix, K are structural stiffness matrix and B_cFor controlling the positional matrix of power u.The vector acceleration of x respectively structure, velocity vector and motion vector. Definition status variableThen dynamics equations (1) can be rewritten as following state space form:

$\left[\begin{array}{c}\stackrel{\·\·}{x}\\ \stackrel{\·}{x}\end{array}\right]=\left[\begin{array}{cc}-M^{-1}P& -M^{-1}K\\ I& 0\end{array}\right]\left[\begin{array}{c}\stackrel{\·}{x}\\ x\end{array}\right]+\left[\begin{array}{c}{M}^{-1}{B}_c\\ 0\end{array}\right]u---\left(2\right)$

That is:

$\begin{array}{c}\stackrel{\·}{X}=AX+BU\\ Y=CX+DU\end{array}---\left(3\right)$

Wherein:C is for extracting matrix, and D normal conditions are null matrix.

Second step: after utilizing the first step to obtain the state space form of active control system, needs to carry out the uncertainty analysis of active control system below. Owing to structural parameters exist uncertainty, the state-space model thus resulting in structural control system is also probabilistic, say, that the parameter of state space equation (3) is uncertain parameter, it may be assumed that

$A\∈\[\underset{\‾}{A},\stackrel{\‾}{A}\],B\∈\[\underset{\‾}{B},\stackrel{\‾}{B}\]---\left(4\right)$

Here it is left out extracting the uncertainty of Matrix C. WhereinA,The respectively lower bound of matrix A and the upper bound.B,The respectively lower bound of matrix B and the upper bound. Utilize bounded-but-unknown uncertainty to analyze method and can obtain the bound of state variable X, be namely interval boundary:

$X\∈\[\underset{\‾}{X},\stackrel{\‾}{X}\]---\left(5\right)$

3rd step: obtain the interval boundary of the state variable of closed-loop control system in second step, it is possible to utilize these interval boundaries that vibration active control system is carried out Multidisciplinary systems tolerance. Utilize the Multidisciplinary systems metric computational methods of invention, closed loop active control system is carried out fail-safe analysis. When marginal value is the real number determined, it is possible to use following computing formula carries out non-probability decision degree and calculates

$Pos\left(sys\right)=\left\{\begin{array}{cc}0& if{X}_{cri}<\underset{\&OverBar;}{X}\\ \frac{X_{cri}-\underset{\&OverBar;}{X}}{\stackrel{\&OverBar;}{X}-\underset{\&OverBar;}{X}}& if\underset{\&OverBar;}{X}<X_{cri}<\stackrel{\&OverBar;}{X}\\ 1& if{X}_{cri}=\stackrel{\&OverBar;}{X}\\ 1+X_{cri}-\stackrel{\&OverBar;}{X}& if\stackrel{\&OverBar;}{X}<X_{cri}\end{array}\right.---\left(6\right)$

The non-probability decision degree that wherein Pos (sys) is system.X_criFor the marginal value of response, this value is given by designer and design, can determine according to practical problem and control requirement.

4th step: the weighting function in Optimal Control Problem Solution process is optimized. Reliability optimization model is as follows:

$\begin{array}{c}findQ,R\\ \min \max \left(u\left(t\right)\right)\\ s.t.Pos\left(sys\right)\&GreaterEqual;R_{cri}\end{array}---\left(7\right)$

Wherein: Q, R are the weighting functions in optimum control, also it is the design variable of Optimized model. Max (u (t)) is the maximum controlling input power, represents that object function is to, under meeting the constraint that reliability requires, make output control power minimum as far as possible with it. The non-probability decision degree that Pos (sys) is system can be tried to achieve by formula (6). R_criFor designer require reliability, be a set-point. In order to meet the control minimum R of power output_criGenerally it is taken as 1. Tradition this reliability value of robust control will be generally above 1, which results in the conservative of system, be need control power output bigger.

5th step: utilize the weighting function after optimizing to carry out solving of optimal controller, design closed loop Active Vibration Control System.

Present invention advantage compared with prior art is in that:

(1) present invention carries out under non-probabilistic framework, it is met the weight function matrix that RELIABILITY DESIGN requires by Multidisciplinary systems optimization, this weighting function is utilized to carry out controller design so that vibration active control system disclosure satisfy that control requirement under condition of uncertainty.

(2) present invention proposes the Multidisciplinary systems index calculating method of active control system. The method can obtain the active control system reliability index in any situation, and the fail-safe analysis for active control system provides the foundation. Also theoretical basis has been established for further Design of reliable controller.

Accompanying drawing explanation

Fig. 1 is that cantilever beam closed loop control controls schematic diagram;

The change in displacement schematic diagram of free end when Fig. 2 is do not add controller;

The change in displacement schematic diagram of free end when Fig. 3 is apply controller;

Fig. 4 is the comparison schematic diagram of system response before and after controlling;

Fig. 5 is tradition optimal control results schematic diagram;

Fig. 6 is reliable optimal control results schematic diagram;

Fig. 7 is tradition optimum control and reliable optimum control control power change schematic diagram;

Fig. 8 is the situation schematic diagram that reliability is equal to 0;

Fig. 9 is interference situation schematic diagram;

Figure 10 is the reliability situation schematic diagram more than 1;

Figure 11 is the comparison schematic diagram of two interval numbers;

Figure 12 is the flowchart of the present invention.

Detailed description of the invention

Below in conjunction with accompanying drawing the present invention is described in further detail embodiments of the present invention.

The present invention is applicable to the active control in structural vibration problem under non-probabilistic framework. In field of active control in structural vibration, generally requiring in the face of various uncertain problems, uncertainty tends to affect the control effect of active control system, and What is more is likely to the stability of destruction system. In order to solve the uncertain problem faced in active Vibration Control Design process, weighting function in optimum control is optimized by the present invention based on Multidisciplinary systems optimization method, acquisition meets design requirement to obtain controller, and final design goes out reliable optimum closed-loop control system. This system exists in parameter and still can meet design requirement in probabilistic situation, and can be effectively prevented from the problem that robust control is overly conservative.

The present invention is first according to modern control theory, the state space form of structure of having derived vibration equation, the active control system Multidisciplinary systems being then based on proposing analyzes method, the uncertain Interval promulgation algorithm of bonding state space equation, carries out uncertainty analysis and reliability calculating to uncertain active control system. Utilizing optimized algorithm to obtain the weighting function of optimum, finally design obtains reliable optimal control system, and as shown in figure 12, implementation step is as follows:

The first step: according to Structural Dynamics finite element equation, set up the state space equation of active control system:

$M\stackrel{\&CenterDot;\&CenterDot;}{x}+P\stackrel{\&CenterDot;}{x}+Kx=B_{c}u---\left(1\right)$

Wherein M to be architecture quality matrix, P be structural damping matrix, K are structural stiffness matrix and B_cFor controlling the positional matrix of power u.The vector acceleration of x respectively structure, velocity vector and motion vector. Definition status variableThen dynamics equations (1) can be rewritten as following state space form:

$\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{x}\\ \stackrel{\&CenterDot;}{x}\end{array}\right]=\left[\begin{array}{cc}-M^{-1}P& -M^{-1}K\\ I& 0\end{array}\right]\left[\begin{array}{c}\stackrel{\&CenterDot;}{x}\\ x\end{array}\right]+\left[\begin{array}{c}{M}^{-1}{B}_c\\ 0\end{array}\right]u---\left(2\right)$

That is:

$\begin{array}{c}\stackrel{\&CenterDot;}{X}=AX+BU\\ Y=CX+DU\end{array}---\left(3\right)$

Wherein:C is for extracting matrix, and D normal conditions are null matrix.

Second step: the uncertainty analysis of active control system. Owing to structural parameters exist uncertainty, the state-space model thus resulting in structural control system is also probabilistic, say, that the parameter of state space equation (3) is uncertain parameter, it may be assumed that

$A\&Element;\&lsqb;\underset{\&OverBar;}{A},\stackrel{\&OverBar;}{A}\&rsqb;,B\&Element;\&lsqb;\underset{\&OverBar;}{B},\stackrel{\&OverBar;}{B}\&rsqb;---\left(4\right)$

Here it is left out extracting the uncertainty of Matrix C. WhereinA,The respectively lower bound of matrix A and the upper bound.B,The respectively lower bound of matrix B and the upper bound. Utilize bounded-but-unknown uncertainty to analyze method and can obtain the bound of state variable X, namely

$X\&Element;\&lsqb;\underset{\&OverBar;}{X},\stackrel{\&OverBar;}{X}\&rsqb;---\left(5\right)$

Owing to structural system exists uncertain parameterBeing to rely on uncertain parameter b by the mass matrix of the known controlled structure of finite element analysis, damping matrix and stiffness matrix, therefore state space equation (3) can be written as following form:

$\begin{array}{c}\stackrel{\&CenterDot;}{X}\left(b\right)=A\left(b\right)X\left(b\right)+B\left(b\right)U\\ Y\left(b\right)=CX\left(b\right)+DU\end{array}---\left(6\right)$

All states of putative structure all can be surveyed, then the dynamic response of structure can carry out utilization state feedback controller actively control, it may be assumed that

U=-GX (7)

Wherein G is by being tried to achieve controller.

Equation (7) is brought in equation (6) and can be obtained by the closed loop active control system containing interval parameter:

$\begin{array}{c}\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\left(b\right)-B\left(b\right)G\right)X\left(b\right)\\ Y\left(b\right)=CX\left(b\right)+DU\end{array}---\left(8\right)$

Its reliability could be measured after obtaining the interval of response of closed-loop control system. In order to analyze the reliability of closed-loop control system, first have to how clearly uncertainty is propagated in closed-loop control system. When parameter b changes, the solution meeting equation (8) has infinite multiple, and these solve response sets composed as follows

$\mathrm{\&Gamma;}=\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\right(b)-B(b\left)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\}---\left(9\right)$

In general, set Γ be one be difficult to obtain, extremely complex region, it is possible to transfer to and solve the border finding response sets.

$\begin{array}{c}\underset{\&OverBar;}{X}=\max \left\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\left(b\right)-B\left(b\right)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\right\}\\ \stackrel{\&OverBar;}{X}=\max \left\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\left(b\right)-B\left(b\right)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\right\}\end{array}---\left(10\right)$

Utilize Taylor series expansion and interval extension computing, it is possible to obtain the approximate solution of equation (10)

$\begin{array}{c}\underset{\&OverBar;}{X}=X\left(b^c\right)-\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\left|\frac{\&part;X\left(b^c\right)}{\&part;b_j}\right|{\mathrm{\&Delta;b}}_j\\ \stackrel{\&OverBar;}{X}=X\left(b^c\right)+\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\left|\frac{\&part;X\left(b^c\right)}{\&part;b_j}\right|{\mathrm{\&Delta;b}}_j\end{array}---\left(11\right)$

Wherein: b^cFor interval b^IAverage, Δ b_jFor variable b_jRadius, m is the number of uncertain variables.

3rd step: vibration active control system Multidisciplinary systems is measured. Through setting up state space equation and uncertainty analysis, have been obtained for closed-loop control system response interval. Utilize the Multidisciplinary systems metric computational methods of invention, closed loop active control system is carried out fail-safe analysis. When marginal value is the real number determined, it is possible to use following computing formula carries out non-probability decision degree and calculates:

$Pos\left(sys\right)=\left\{\begin{array}{cc}0& if{X}_{cri}<\underset{\&OverBar;}{X}\\ \frac{X_{cri}-\underset{\&OverBar;}{X}}{\stackrel{\&OverBar;}{X}-\underset{\&OverBar;}{X}}& if\underset{\&OverBar;}{X}<X_{cri}<\stackrel{\&OverBar;}{X}\\ 1& if{X}_{cri}=\stackrel{\&OverBar;}{X}\\ 1+X_{cri}-\stackrel{\&OverBar;}{X}& if\stackrel{\&OverBar;}{X}<X_{cri}\end{array}\right.$

The non-probability decision degree that wherein Pos (sys) is system. X_criMarginal value for response.

(1) when the response interval of closed loop active control system comprises marginal value: when closed-loop control system response quautity is less than given marginal value, namely closed-loop control system is safe (reliably), closed-loop control system response quautity can be defined and be called the non-probability decision degree of closed loop active control system less than the uncertain proposition of given marginal value, it is defined as security domain area and the enclosed area ratio of uncertain variables, namely

P_s=Prob (d < d^cri)=S_safe/S_total

The uncertain proposition that corresponding closed loop active control system lost efficacy can be defined as inefficacy territory area and the enclosed area ratio of uncertain variables, it may be assumed that

P_s=Prob (d < d^cri)=(S_total-S_failure)/S_total

If the interval variable that limit state equation contains certainly is more than two, then interval variable area defined is cuboid or hypercube, and now, non-probability decision degree can be defined as the hypervolume of safety zone and the ratio of cumulative volume. Certainly, if limit state equation has nonlinear form, the definition method of this non-probability decision degree is also applicable. When marginal value be a certain determine several time, Multidisciplinary systems model is just degenerated to length ratio on number axis.

(2) when the response interval of closed loop active control system does not comprise marginal value: be now divided into again two kinds of situations, a) lower bound that marginal value responds less than closed loop active control system.Now the reliability of active control system is 0. B) upper bound that marginal value responds more than closed loop active control system. Now the reliability of active control system is more than 1, and occurrence can utilize formula (12) to obtain.

4th step: the weighting function in Optimal Control Problem Solution process is optimized. Reliability optimization model is as follows:

$\begin{array}{c}findQ,R\\ \min \max \left(u\left(t\right)\right)\\ s.t.Pos\left(sys\right)\&GreaterEqual;R_{cri}\end{array}---\left(7\right)$

Wherein: Q, R are the weighting functions in optimum control, also it is the design variable of Optimized model. Max (u (t)) is the maximum controlling input power, represents that object function is to, under meeting the constraint that reliability requires, make output control power minimum as far as possible with it. The non-probability decision degree of Pos (sys) system can be tried to achieve by formula (6). R_criFor designer require reliability, be a set-point, in order to meet control the minimum R of power output_criGenerally it is taken as 1. Tradition this reliability value of robust control will be generally above 1, which results in the conservative of system, be need control power output bigger.

(1) first weighting function Q, R are carried out parametrization, given initial weighting function Q, R.

(2) under given weighting function, the theory of optimal control is utilized to obtain optimal controller.

(3) the controller design closed-loop control system tried to achieve is utilized.

(4) analyze the reliability of closed-loop control system, if meet design requirement, otherwise return (1). Satisfied then continue.

(5) the control power of computing controller output, if for Optimal Control Force (minimum), otherwise return (1). Satisfied then continue.

(6) reliable optimal controller is obtained, reliable design optimum closed-loop control system.

5th step: utilize the weighting function after optimizing to carry out solving of optimal controller, utilizes MATLAB to control workbox and can be obtained by controller, utilize the controller obtained just can build closed-loop control system.

Specific embodiment is as follows:

Considering cantilever beam structure as shown in Figure 1, cantilever beam is freely being subject to initial disturbance, and displacement is 5mm, utilizes invention design con-trol device so that the degree of freedom of cantilever beam after 0.05s maximum displacement less than 1.1mm. Driver is positioned at the center of cantilever beam, ten sensors are positioned at below cantilever beam, and by observer, all states of cantilever beam are all estimated, being then delivered to controller, controller is applied to the upper surface center of cantilever beam through computing place of production voltage signal. Physical dimension and the material properties of cantilever beam structure are as shown in table 1.

The physical dimension of table 1 cantilever beam structure and material properties

The first step: cantilever beam structure is carried out finite element modeling first with large commercial finite element analysis software ANSYS. Utilize the following mass matrix of order derived type structure, stiffness matrix:

/ solu enters and solves module

Antype, 7! Substructuring selects to solve module

Seopt, matname, 3 be structural information everywhere

Nsel, all select all nodes

M, all, all select all nodes to be master unit

Solve solves

Selist, matname, 3 list all unit and node information

Definition status variableThen dynamics equations can be rewritten as following state space form:

$\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{x}\\ \stackrel{\&CenterDot;}{x}\end{array}\right]=\left[\begin{array}{cc}-M^{-1}P& -M^{-1}K\\ I& 0\end{array}\right]\left[\begin{array}{c}\stackrel{\&CenterDot;}{x}\\ x\end{array}\right]+\left[\begin{array}{c}{M}^{-1}{B}_c\\ 0\end{array}\right]u$

That is:

$\stackrel{\&CenterDot;}{X}=AX+BU$

Y=CX+DU

Wherein:C is for extracting matrix, and D is null matrix.

Second step: the analytical structure structure dynamic response when not applying control power. Utilize equation below or utilize finite element analysis software can obtain the vibration displacement curve of cantilever beam free end as shown in Figure 2.

$x\left(t\right)=\exp \left(At\right)x\left(0\right)+{\&Integral;}_0^t\exp \&lsqb;A(t-\mathrm{\&tau;})\&rsqb;Bu\left(\mathrm{\&tau;}\right)d\mathrm{\&tau;}$

X (t) is system state variables, and exp is natural number e.

3rd step: the uncertainty analysis of cantilever beam active control system. Owing to structural parameters exist uncertainty, the state-space model thus resulting in structural control system is also probabilistic, say, that the parameter of state space equation is uncertain parameter, it may be assumed that

$A\&Element;\&lsqb;\underset{\&OverBar;}{A},\stackrel{\&OverBar;}{A}\&rsqb;,B\&Element;\&lsqb;\underset{\&OverBar;}{B},\stackrel{\&OverBar;}{B}\&rsqb;$

The present embodiment definition Δ A=A^c× 5%, Δ B=B^c× 5%. Here it is left out extracting the uncertainty of Matrix C. WhereinA,The respectively lower bound of matrix A and the upper bound.B,The respectively lower bound of matrix B and the upper bound. Utilize bounded-but-unknown uncertainty to analyze method and can obtain the bound of state variable X, it may be assumed that

$X\&Element;\&lsqb;\underset{\&OverBar;}{X},\stackrel{\&OverBar;}{X}\&rsqb;$

Owing to structural system exists uncertain parameterBeing to rely on uncertain parameter b by the mass matrix of the known controlled structure of finite element analysis, damping matrix and stiffness matrix, therefore state space equation can be written as following form:

$\stackrel{\&CenterDot;}{X}\left(b\right)=A\left(b\right)X\left(b\right)+B\left(b\right)U$

Y (b)=CX (b)+DU

All states of putative structure all can be surveyed, then the dynamic response of structure can carry out utilization state feedback controller actively control, it may be assumed that

U=-GX

Wherein G is by being tried to achieve controller. Obtain the closed loop active control system containing interval parameter:

$\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\right(b)-B(b\left)G\right)X\left(b\right)$

Y (b)=CX (b)+DU

Its reliability could be measured after obtaining the interval of response of closed-loop control system. In order to analyze the reliability of closed-loop control system, first have to how clearly uncertainty is propagated in closed-loop control system. When parameter b changes, meeting non trivial solution has infinite multiple, and these solve response sets composed as follows:

$\mathrm{\&Gamma;}=\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\right(b)-B(b\left)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\}$

In general, set Γ be one be difficult to obtain, extremely complex region, it is possible to transfer to and solve the border finding response sets.

$\underset{\&OverBar;}{X}=max\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\right(b)-B(b\left)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\}$

$\stackrel{\&OverBar;}{X}=max\{X\left(b\right):\stackrel{\&CenterDot;}{X}\left(b\right)=\left(A\right(b)-B(b\left)G\right)X\left(b\right),b\&Element;\&lsqb;\underset{\&OverBar;}{b},\stackrel{\&OverBar;}{b}\&rsqb;=b^I\}$

Utilize Taylor series expansion and interval extension computing, it is possible to obtain the approximate solution of equation:

$\underset{\&OverBar;}{X}=X\left(b^c\right)-\underset{j=1}{\overset{m}{\&Sigma;}}\left|\frac{\&part;X\left(b^c\right)}{\&part;b_j}\right|{\mathrm{\&Delta;b}}_j$

$\stackrel{\&OverBar;}{X}=X\left(b^c\right)+\underset{j=1}{\overset{m}{\&Sigma;}}\left|\frac{\&part;X\left(b^c\right)}{\&part;b_j}\right|{\mathrm{\&Delta;b}}_j$

Wherein: b^cFor interval b^IAverage, Δ b_jFor variable b_jRadius, m is the number of uncertain variables. The present embodiment we first define Q=0.01I_120×120, R=1 × 10⁵I_1×1, wherein I_120×120It is the unit matrix of 120 × 120, I_1×1It is 1. Obtain closed-loop control system. Then the closed-loop control system obtained with the present invention contrasts. Fig. 5 gives upper and lower bounds of responses when traditional closed-loop controls.

4th step: vibration active control system Multidisciplinary systems is measured. Through modeling and uncertainty analysis, have been obtained for closed-loop control system response interval. Utilize the Multidisciplinary systems metric method of invention, closed loop active control system is carried out fail-safe analysis. When marginal value is the real number determined, it is possible to use following computing formula carries out non-probability decision degree and calculates:

$Pos\left(sys\right)=\left\{\begin{array}{cc}0& if{X}_{cri}<\underset{\&OverBar;}{X}\\ \frac{X_{cri}-\underset{\&OverBar;}{X}}{\stackrel{\&OverBar;}{X}-\underset{\&OverBar;}{X}}& if\underset{\&OverBar;}{X}<X_{cri}<\stackrel{\&OverBar;}{X}\\ 1& if{X}_{cri}=\stackrel{\&OverBar;}{X}\\ 1+X_{cri}-\stackrel{\&OverBar;}{X}& if\stackrel{\&OverBar;}{X}<X_{cri}\end{array}\right.$

The non-probability decision degree that wherein Pos (sys) is system. X_criMarginal value for response.

(1) when the response interval of closed loop active control system comprises marginal value: when closed-loop control system response quautity is less than given marginal value, namely closed-loop control system is safe (reliably), closed-loop control system response quautity can be defined and be called the non-probability decision degree of closed loop active control system less than the uncertain proposition of given marginal value, it is defined as security domain area and the enclosed area ratio of uncertain variables such as shown in Fig. 9 or Figure 11, namely

P_s=Poss (d < d^cri)=S_safe/S_total

Wherein P_sFor non-probability decision degree, S_safeFor the area of security domain, S_totalThe gross area that uncertain variables surrounds. D is the real response of system, d^criFor the marginal value that system response was lost efficacy.

The uncertain proposition that corresponding closed loop active control system lost efficacy can be defined as inefficacy territory area and the enclosed area ratio of uncertain variables, it may be assumed that

P_s=Prob (d < d^cri)=(S_total-S_failure)/S_total

If the interval variable that limit state equation contains certainly is more than two, then interval variable area defined is cuboid or hypercube, and now, non-probability decision degree can be defined as the hypervolume of safety zone and the ratio of cumulative volume.Certainly, if limit state equation has nonlinear form, the definition method of this non-probability decision degree is also applicable. When marginal value be a certain determine several time, Multidisciplinary systems model is just degenerated on number axis length ratio as shown in Figure 9.

(2) when the response interval of closed loop active control system does not comprise marginal value: be now divided into again two kinds of situations, a) lower bound that marginal value responds less than closed loop active control system. Now the reliability of active control system is 0, as shown in Figure 8. B) upper bound that marginal value responds more than closed loop active control system. Now the reliability of active control system is more than 1, and occurrence can utilize formula to obtain as shown in Figure 10, and wherein, d is assumed to the response output of system, d^cFor the nominal value of system, d^criRepresent the marginal value of design.dWithIt is assumed to lower bound and the upper bound of response respectively,d ^criWithThe respectively lower bound of marginal value and the upper bound.

The comparison of table 2 present invention and traditional method

5th step: the weighting function in Optimal Control Problem Solution process is optimized as shown in figure 12. Reliability optimization model is as follows:

findQ,R

minmax(u(t))

s.t.Pos(sys)≥R_cri

Wherein: Q, R are the weighting functions in optimum control, also it is the design variable of Optimized model. Max (u (t)) is the maximum controlling input power, represents that object function is to, under meeting the constraint that reliability requires, make output control power minimum as far as possible with it. The non-probability decision degree of Pos (sys) system can be tried to achieve by formula (6). R_criFor the reliability that designer requires, in order to meet control, power output is minimum generally takes R_criIt is 1. Tradition this reliability value of robust control will be generally above 1, which results in the conservative of system, be need control power output bigger.

(1) first weighting function Q, R are carried out parametrization, given initial weighting function Q, R.

(2) under given weighting function, the theory of optimal control is utilized to obtain optimal controller.

(3) the controller design closed-loop control system tried to achieve is utilized.

(4) analyze the reliability of closed-loop control system, if meet design requirement, otherwise return (1). Satisfied then continue.

(5) the control power of computing controller output, if for Optimal Control Force (minimum), otherwise return (1). Satisfied then continue.

(6) reliable optimal controller is obtained, reliable design optimum closed-loop control system.

Eventually pass that to optimize the weighting function that obtains be Q=0.01I_120×120, R=1 × 10⁵I_1×1, utilize this group weighting function design optimal controller, meet reliability requirement.

6th step: design closed loop Active Vibration Control System. Fig. 3 gives the displacement curve of the cantilever beam degree of freedom after control, and Fig. 4 gives the contrast before and after controlling, it can be seen that increases controller and has reached the effect of control structure vibration. It is 100% reliably that Fig. 6 gives reliable optimal controller, will not lose efficacy. Although traditional controller average meets design requirement, but when uncertainty exists, the response of system has the probability of 1-87.12% to lose efficacy. Meanwhile, Fig. 7 give also Traditional control and the curve of the control power wanted required for the present invention.

Claims (7)

1. the vibration optimal control system method for designing optimized based on Multidisciplinary systems, it is characterised in that step is as follows:

The first step: according to Structural Dynamics finite element equation, set up the state space equation of active control system;

Second step: carry out the uncertainty analysis of active control system on the basis of the first step, utilizes bounded-but-unknown uncertainty to analyze method and obtains the bound of state variable X, namely

3rd step: vibration active control system Multidisciplinary systems is measured, uncertainty analysis through the state space equation of active control system of the first step and second step, obtain closed-loop control system response interval, utilize the computational methods of Multidisciplinary systems metric, closed loop active control system is carried out fail-safe analysis, calculates and obtain closed loop active control system non-probability decision degree Pos (sys);

4th step: the weighting function in Optimal Control Problem Solution process is optimized, on the basis of calculated non-probability decision degree Pos (sys), is optimized the weighting function in Optimal Control Problem Solution process, weighting function Q, the R after being optimized; Optimization aim is control the maximum max (u (t)) of input power, represents that object function is to, under meeting the constraint that reliability requires, make output control power minimum as far as possible with maximum;

5th step: utilizing the weighting function after optimizing to carry out solving of optimal controller, design obtains closed loop Active Vibration Control System.

2. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 1, it is characterized in that: the maximum that step 4 is controlled input power is defined, the control power making control system is minimum, and the reliability of closed-loop control system is maximum.

3. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 1, it is characterised in that: the state space equation of described first step active control system:

$M\stackrel{\&CenterDot;\&CenterDot;}{x}+P\stackrel{\&CenterDot;}{x}+Kx=B_{c}u$

Wherein M to be architecture quality matrix, P be structural damping matrix, K are structural stiffness matrix and B_cFor controlling the positional matrix of power u,The vector acceleration of x respectively structure, velocity vector and motion vector, definition status variableThen state space equation is rewritten as following state space form:

$\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{x}\\ \stackrel{\&CenterDot;}{x}\end{array}\right]=\left[\begin{array}{cc}-M^{-1}P& -M^{-1}K\\ I& 0\end{array}\right]\left[\begin{array}{c}\stackrel{\&CenterDot;}{x}\\ x\end{array}\right]+\left[\begin{array}{c}{M}^{-1}{B}_c\\ 0\end{array}\right]u$

That is:

$\stackrel{\&CenterDot;}{X}=AX+BU$

Y=CX+DU

Wherein: $A=\left[\begin{array}{cc}-M^{-1}P& -M^{-1}K\\ I& 0\end{array}\right],B=\left[\begin{array}{c}{M}^{-1}{B}_c\\ 0\end{array}\right],$ C is for extracting matrix, and D is null matrix.

4. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 1, it is characterised in that: in described 3rd step, the computational methods of Multidisciplinary systems metric:

When marginal value is the real number determined, utilizes following computing formula to carry out non-probability decision degree and calculate:

$Pos\left(sys\right)=\left\{\begin{array}{ccc}0& if& X_{cri}<\underset{\&OverBar;}{X}\\ \frac{X_{cri}-\underset{\&OverBar;}{X}}{\stackrel{\&OverBar;}{X}-\underset{\&OverBar;}{X}}& if& \underset{\&OverBar;}{X}<X_{cri}<\stackrel{\&OverBar;}{X}\\ 1& if& X_{cri}=\stackrel{\&OverBar;}{X}\\ 1+X_{cri}-\stackrel{\&OverBar;}{X}& if& \stackrel{\&OverBar;}{X}<X_{cri}\end{array}\right.$

Wherein Pos (sys) is non-probability decision degree, X_criMarginal value for response.

5. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 1, it is characterised in that: when the weighting function in Optimal Control Problem Solution process being optimized in described 4th step, reliability optimization model is as follows:

findQ,R

minmax(u(t))

s.t.Pos(sys)≥R_cri

Wherein: Q, R are the weighting functions in optimum control, also it is the design variable of Optimized model; Max (u (t)) is the maximum controlling input power, and Pos (sys) is non-probability decision degree; R_criFor designer require reliability, for set-point.

6. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 5, it is characterised in that: minimum in order to meet control power output, R_criIt is taken as 1.

7. the vibration optimal control system method for designing optimized based on Multidisciplinary systems according to claim 1, it is characterised in that: utilizing the weighting function after optimizing to carry out solving of optimal controller, it is as follows that design obtains closed loop Active Vibration Control System process:

(1) utilize the state space equation that the first step obtains, Matlab/Simulink sets up the state-space model of response;

(2) according to obtain weighting function utilize in Matlab optimum control case design obtain optimal controller;

(3) in Matlab/Simulink, utilize the optimal controller assembly feedback control system obtained in (2).